In 1902 Burnside asked a question that was to become very influential in group theory: Can a group with all elements of finite order be infinite?

This question has found a positive answer in 1964 with the work of Golod and Shafarevich, but also in 1980 with a group generated by an automaton with only 5 states and 2 letters. Then, since the 80’s, automaton groups arise as a powerful source of example of interesting groups.

In this talk we will discuss the impact of the structural properties of the underlying automaton on the generated group and present a combinatorial tool, the orbit tree of an automaton, that can be used to prove that some classes of automata cannot generate infinite Burnside groups. We will also give some idea of the proofs. No prior background is required.